Multiple Criteria Decision Making and General Regression for Determining Influential Factors on S & P 500 Index Futures

We employ the DEMATEL-based analytic network process (D-ANP) to evaluate the weight of various factors on S&P 500 index futures. The general regression method is employed to prove the result. We then employed grey relational analysis (GRA) to examine predictive power of determinants suggested by 13 experts for fluctuations in S&P 500 index futures. This study yields a number of empirical results. (1) The explanatory power of macroeconomic factors for S&P 500 index futures outperforms that of technical indicators, as found in most of previous research papers; (2) The D-ANP revealed that five core factors (US dollar index, ISM manufacturing purchasing managers’ index (PMI), interest rate, volatility index, and unemployment rate) affect fluctuations in S&P 500 index futures, of which the US dollar index is the most important; (3) A casual diagram shows that the US dollar index and interest rate have mutual effects, and the US dollar index unilaterally affects ISM manufacturing PMI, unemployment rate, and the volatility index; (4) Granger causality test results confirmed some similar results obtained via the D-ANP that the US dollar index, interest rate, and the PMI have major impacts on the S&P 500 index futures; (5) The general regression results confirmed that four of five factors selected via the D-ANP (US dollar index, interest rate, volatility index, and unemployment rate) have strong explanatory power in forecasting the rate of return on S&P 500 index futures; (6) The GRA revealed that the explanatory power of various factors selected via the D-ANP was better for S&P 500 than for Dow Jones Industrial Average (DJIA) and Nasdaq 100 index futures; (7) The explanatory power is better for S&P 500 Industrial than for S&P 500 transportation, utility, and financial index futures.


Introduction
Over the past two decades, international financial markets fluctuated dramatically because of the US subprime loan crisis and five European countries debt crises.To prevent shocks induced by huge volatility in stock fluctuations in the near future, investors are anxious for identifying appropriate hedging instruments.The Taiwanese government approved listings of the ETFs of S&P 500, Nasdaq, and Dow Jones industrial indexes on the Taiwan Stock Exchange since December, 2015.These listings not only connect Taiwanese stock markets with international markets, but also provide Taiwanese investors with international financial instruments.
With the closed relationship among the financial markets of various countries, this study emphasizes that, in addition to technical factors, macroeconomic factors of each country play an Symmetry 2018, 10, 5 2 of 25 essential role.In fact, the trend for US stocks has strongly and continuously rebounded since the fourth quarter of 2015 to that of 2017, and the financial reports of enterprises are much better than expected, so investor prospects have changed from extreme pessimism to extreme optimism.Since a combination of Decision Making Trial and Evaluation Laboratory (DEMATEL) and analytic network process (ANP) has been widely used to solve various decision problems considering interdependencies among factors [1], we use the DEMATEL-based ANP (D-ANP) to examine impacts among factors that can affect investors trade in S&P 500 index futures.We also perform the general regression method to confirm the results obtained by the D-ANP.Furthermore, we examine the explanatory power for four major sectors of the S&P 500 index and for three major US stock index futures for various factors selected via the D-ANP.The grey relational analysis (GRA) [2] was further to evaluate S&P 500 sectors.
The study objectives are as follows: (1) to pick various key factors out of 19 factors affecting investors trading in S&P 500 index futures by 13 experts via multiple-round questionnaires; (2) to examine mutual relationships among various key factors affecting investor trading in S&P 500 index futures using Delphi and D-ANP with the Borda method; (3) to examine the causal relationship among the five factors selected by the D-ANP, then use Granger causality test and the general regression method to investigate the relationship; (4) to examine the explanatory power for four different S&P 500 index sectors using the GRA; and (5) to examine the explanatory power for three major U.S. stock indexes futures using the GRA for various factors selected via the D-ANP.
It is known that, it is too time-consuming if there are various criteria regarding pairwise comparisons.The DEMATEL method was then employed for solving complicated problems, and a causal diagram was used for policy-making and for exploratory, theoretical, and large-scale empirical studies.DEMATEL was also employed to solve inner dependency problems among a set of criteria [12,13].The D-ANP was used to solve the problems with ANP due to pairwise comparisons [1,[14][15][16][17][18][19][20][21].

Delphi Method
We first selected 19 factors that might affect S&P 500 index futures from our literature review.We then used the Delphi method to identify cause-effect relationships and weights for factors affecting S&P 500 index futures.

Invite Qualified Experts
Gordon and Helmer [38] proposed the Delphi method which used a continuous series of questionnaires to draw out predictions from various experts in many rounds.After each round, the result was sent back to each expert to make some adjustments based on the viewpoints from other experts.Finally, the process was completed after the accomplishment of consistency, and the average score from the final round was calculated.In this study, we invited 13 experts who had been working in the relevant field for more than five years.The academic and professional background for 13 experts are presented at

Prototypical Structure
We first selected 19 factors affecting S&P 500 investment strategy from previous research papers and then used the Delphi method to interview 13 experts to develop a prototypical structure, as shown in Figure 1.

Prototypical Structure
We first selected 19 factors affecting S&P 500 investment strategy from previous research papers and then used the Delphi method to interview 13 experts to develop a prototypical structure, as shown in Figure 1.

Amendment of the Prototypical Structure
Table 2 shows how the prototypical structure was amended using the responses of 13 experts to the prototypical structure.Six factors with a weak relationship with the research subject were deleted.According to the amended prototypical structure, we developed a first-round questionnaire and asked 13 experts to rank each factor using a score ranging from 0 to 100.To examine the consistency among the experts, we used the consensus deviation index (CDI) to check the accuracy and set the threshold to 0.1.Table 3 shows the average CDI score and standard deviation for the first-round questionnaire.
Figure 2 shows the amendment of the prototypical structure according to the suggestions of 13 experts.Table 2 shows how the prototypical structure was amended using the responses of 13 experts to the prototypical structure.Six factors with a weak relationship with the research subject were deleted.According to the amended prototypical structure, we developed a first-round questionnaire and asked 13 experts to rank each factor using a score ranging from 0 to 100.To examine the consistency among the experts, we used the consensus deviation index (CDI) to check the accuracy and set the threshold to 0.1.Table 3 shows the average CDI score and standard deviation for the first-round questionnaire.
Figure 2 shows the amendment of the prototypical structure according to the suggestions of 13 experts.3 indicates that the CDI was >0.1 for eight of 13 factors, suggesting that the 13 experts did not have a consensus viewpoint for the first-round questionnaire.Therefore, we conducted a second-round questionnaire asking the experts to revise their first answers.The findings are presented at Table 4.
Table 4 indicates that, after the second-round questionnaire, the CDI was <0.1 for all 13 factors, suggesting that all 13 experts have consensus opinions on the second-round questionnaire.We then rearranged the order according to the average score given by the experts.Since all the experts agreed that an average score of 50 was the threshold, we deleted factor A8 as its mean score was only 48.85. Figure 3 shows the final 12 factors retained in the formal research structure.
Symmetry 2018, 10, x FOR PEER REVIEW 6 of 25 3.1.5.Results for the Second-Round Questionnaire Table 3 indicates that the CDI was >0.1 for eight of 13 factors, suggesting that the 13 experts did not have a consensus viewpoint for the first-round questionnaire.Therefore, we conducted a secondround questionnaire asking the experts to revise their first answers.The findings are presented at Table 4.
Table 4 indicates that, after the second-round questionnaire, the CDI was <0.1 for all 13 factors, suggesting that all 13 experts have consensus opinions on the second-round questionnaire.We then rearranged the order according to the average score given by the experts.Since all the experts agreed that an average score of 50 was the threshold, we deleted factor A8 as its mean score was only 48.85. Figure 3 shows the final 12 factors retained in the formal research structure.

D-ANP
The ANP employs pairwise comparisons to judge the weights for the factors of the structure and rank the possible choices in the decision.The ANP consists of the following four major stages [3,39].The advantage of the D-ANP is that it took the total influence matrix generated by DEMATEL as the unweighted supermatrix of ANP directly to avoid troublesome pairwise comparisons [1,19].The flowchart of the D-ANP is depicted in Figure 4 [17].The detail of this flowchart can be referred to [17].In the present study, 13 experts were asked to rank the order of various factors, based on the importance of each factor, using the DEMATEL.

D-ANP
The ANP employs pairwise comparisons to judge the weights for the factors of the structure and rank the possible choices in the decision.The ANP consists of the following four major stages [3,39].The advantage of the D-ANP is that it took the total influence matrix generated by DEMATEL as the unweighted supermatrix of ANP directly to avoid troublesome pairwise comparisons [1,19].The flowchart of the D-ANP is depicted in Figure 4 [17].The detail of this flowchart can be referred to [17].In the present study, 13 experts were asked to rank the order of various factors, based on the importance of each factor, using the DEMATEL.

GRA
GRA is a useful method for evaluating alternatives.Situations for which information is lacking or incomplete are described as being grey.We used GRA to handle similar degree of complicated relations.The main idea of GRA is to obtain a grey relational grade (GRG), which can be used to explain the relationship among relevant factors.The major purpose of GRA is to measure the GRG among factors, so that the crucial rules influencing the development of the system can be found; then, the major performance characteristics of the research target can be grasped [23].To choose the multiple alternatives, every alternative is arranged through data sequence.Any two series have a certain degree of relations [2].
We used GRA to analyze reference and comparative sequences to examine mutual relationship among factors.GRA treats the reference sequence as the goal to achieve, and examines the extent to which the comparative sequence approaches the reference sequence.GRA is an influence assessment model that measures the extent of likeness or unlikeness between two sequences based on the GRG.GRA allows comprehensive comparison between two sets of data rather than partial comparison by determining the length between two points.To retain this strength, all the criteria are assigned to a single level to the decision theorem.GRA was not required to find the best solution, but provides the methods for obtaining right answers for real world problems.
In present study, once the factors were identified by the 13 experts, we measured their performance for four major S&P 500 sectors: (1) industrial; (2) transportation; (3) utility; and (4) financial.We also measured the performance of three major US stock indexes (S&P 500, Nasdaq 100, and DJIA).Evaluations of the four sectors and three US stock indexes were treated as grey system problems because the information is incomplete.

ADF Test
We used an augmented Dickey-Fuller (ADF) model to examine whether a unit root exists at some level of confidence.The ADF model is presented below: where ∆Y t demonstrates the first-order difference of the logarithmic series; α 0 denotes a constant; T represents a time trend; n shows the lag term; δ, γ, and ρ i are the coefficients; and ε t indicates a white noise term in the Hypothesis 1: δ = 0.If one cannot reject the null hypothesis, this suggests that a unit root exists, and it is necessary to take some-order differencing to turn it into a being stationary.

Co-Integration Test and Vector Error Correction Model
We then used co-integration test to examine whether the linear combination of the various variables is stationary.The concept of co-integration can be generalized to schemes of higher-order variables if a linear combination reduces their common order of integration.We employed the maximum likelihood estimation (MLE) model proposed by Johansen and Juselius [40] as follows: where Z t is an endogenous variable of lag p term.
The vector error correction model (VECM) was obtained by employing the first-order differencing from Equation (2), as it adds error correction term (ΠZ t−1 ) to a multi-factor model called vector autoregression (VAR).The VECM is presented as follows: where i=2 A i , p denotes the lagged term, and I represents an identity matrix.
Of which, Π is a long-run influence matrix, and the number of the co-integration vectors is obtained employing the rank of Π matrix.There are three possibilities: (1) Rank (Π) = w, implying that all variables in Z t vectors are stationary time series; (2) Rank (Π) = 0, implying that all variables are stationary time series after performing the first-order difference function, and the variables do not have co-integrating relationship (i.e., they have no long-run equilibrium relationship); (3) Rank (Π) = y, and 0 < y < w, implying that the variables in Z t vectors have y co-integrating relationships.
Based on the Granger's representation theorem, a co-integrated vector can be divided into four parts: a random walk process, a stationary moving average process, a deterministic component, and an item depending on the beginning values, where Π = γδ , of which γ denotes the coefficient matrix of the modifying speed of error correction from non-equilibrium to long-run equilibrium.If γ > 0, suggesting the error of underestimation, then it modifies itself upward by a specific speed to the next period; If γ < 0, indicating the error of overestimation, then it modifies itself downward by a specific speed to the next period.
We used the trace test, which was developed by Johansen and Juselius [40], to estimate all co-integrating vectors, since we have more than two parameters.Trace test proves the wholeness of a witness set of an undeductible variety, allowing for parallel relationship.
Based on the log-likelihood ratio, ln[L max (y)/L max (w)], trace test is performed sequentially for y = w − 1, . . ., 1, 0. This test investigates the null hypothesis that the co-integration rank equals y against the alterative that the rank equals w.The latter implies that Z t is treated stationary.The hypothesis is proposed below: Hypothesis 1. Rank Π ≤ y for the maximum y groups of co-integration vectors.
Hypothesis 2. Rank Π > y for the minimum y groups of co-integration vectors.
The trace test statistics are computed below: where λ trace indicates the statistical value of the trace test; λt represents the estimated value of the ith eigenvalues; T refers to the number of samples; n denotes the number of Eigenvalues that obey the Chi-square distribution under examination.

Granger Causality Test
The Granger causality test is a method to examine causality between two variables in a time series.For a stationary time series, the test is conducted using the exact value of two variables.For a non-stationary time series, the test is conducted employing first (or higher) order difference(s).The number of the lag lengths is usually calculated using an information criterion (i.e., SBC).The Granger causality test deals with two variables, possibly producing incorrect results when the relationship includes more than two parameters.A VAR test will be used when dealing with more than two parameters.
We used the Granger causality test based on the bivariate VAR model as follows: where m 1 and m 2 are intercepts for X t and Y t ; α i and β i indicate the coefficients of the lagged terms of X t and Y t ; γ i and δ i represent the white noises of X t and Y t .Moreover, ε Xt and ε Yt are serially uncorrelated.By employing the F-test, two hypotheses are proposed below: H 0 : There are four cases exist for the causal relationships between X t and Y t : (1) If both hypotheses are rejected, suggesting that X t and Y t are mutually correlated; (2) If H 0 is rejected, indicating that Y t unilaterally affects X t but not vice versa.
(3) If H 0 rather than H 0 is rejected, implying that X t unilaterally affects Y t but not vice versa.(4) If neither hypotheses are rejected, demonstrating that both X t and Y t are independent, implying that X t and Y t do not have causal relationships.

Design of the Third-Round Questionnaire
Using the formal structure as a basis, we applied the D-ANP to carry out third-round questionnaire.Table 5 lists the measurement scores.

D-ANP
The D-ANP was employed in the following stages: Stage 1. Generating a direct impact matrix We generated a direct impact matrix by summarizing the responses from 13 experts.The mean values are presented at Table 6.

Stage 2. Normalizing the direct impact matrix
We added numbers in each row and each column to obtain the maximum value, and the normalized direct impact matrix for 12 factors was presented at Table 7.We added each row to get the dominance effect (d) while adding each column to acquire the reciprocal extent to which a factor is influenced (r); we then calculated the prominence (d + r) and the relation (d − r).
Greater prominence corresponds to greater importance of factors.If the relation was positive, this suggested the factor influenced other factors, and it was therefore defined as a "cause".If the relation Symmetry 2018, 10, 5 13 of 25 was negative, this suggested the factor was influenced by other factors, and it was therefore defined as an "effect".
Table 9 shows that the US dollar index (A2), interest rate (A7), volatility Index (A1), trading volume (B3), and ISM manufacturing PMI (A5) have strong importance for S&P 500 index futures.Stage 5. Generating the weighted supermatrix The total impact matrix from Table 8 was normalized to obtain the weighted supermatrix as presented at Table 10.The limiting supermatrix was determined by multiplying the ANP-weighted supermatrix by itself various times until convergence (refer to Table 11).Using the limiting supermatrix, we calculated the relative weight for each factor, as shown in Table 12.We then calculated the total ranking scores from DEMATEL and ANP methods using Borda's count suggested by Sarri [41] to obtain the final rankings for each factor, as shown in Table 13.The Borda's count is a single-winner election mechanism in which voters rank candidates in order of priority.Since it sometimes elects extensively acceptable candidate instead of those favored by a majority, the Borda's count is usually used as a consensus-based voting mechanism instead of a majoritarian one.
Table 13 reveals that factors A2 and A7 are greatly significant, factors A5, A1, and A3 are very significant, factors A4 and B5 are relatively significant, and factor B1 is insignificant.Hence, the five core factors are A2, A7, A5, A,1 and A3.

Stage 7. Generating a causal diagram for five core factors
A causal diagram for the five core factors was depicted below: Figure 5 depicted that (1) the interest rate and US dollar index are mutually affected; and (2) the US dollar index unilaterally affects the ISM manufacturing PMI, unemployment rate, and volatility index.
Our results suggest that investors should pay attention to the change in interest rates when investing in S&P 500 index futures.core factors are A2, A7, A5, A,1 and A3.
Stage 7. Generating a causal diagram for five core factors A causal diagram for the five core factors was depicted below: Figure 5 depicted that (1) the interest rate and US dollar index are mutually affected; and (2) the US dollar index unilaterally affects the ISM manufacturing PMI, unemployment rate, and volatility index.Our results suggest that investors should pay attention to the change in interest rates when investing in S&P 500 index futures.15 denotes that the skewness of all factors except PMI are positive, suggesting that PMI is skewed left; that means the left tail of PMI is longer than the right side, and the other five factors are skewed to the right.Regarding the kurtosis, we found that S&P 500, interest rate, and unemployment rate are platykurtic distributious (i.e., data distribution with a kurtosis is less than three), and the other three factors are leptokurtic distributions (i.e., data distributions with a kurtosis higher than three).Figure 6.depicts original time series charts for each parameter.

ADF Test
Table 16 denotes that all original data are non-stationary, capable of influencing the behavior of this time series.This first-order difference is taken and all data except unemployment rate under the 1st-order difference column become stationary-order difference for unemployment rate is than taken and unemployment rate under "second-order difference" column of Table 16 become stationary.This result suggests the feasibity of investigating the long-run equilibrium relationship by using the cointegration test [40].

ADF Test
Table 16 denotes that all original data are non-stationary, capable of influencing the behavior of this time series.This first-order difference is taken and all data except unemployment rate under the 1st-order difference column become stationary-order difference for unemployment rate is than taken and unemployment rate under "second-order difference" column of Table 16 become stationary.This result suggests the feasibity of investigating the long-run equilibrium relationship by using the co-integration test [40].Table 19 denotes that unemployment rate was easily modified to the long-run equilibrium with S&P 500 index futures, while the other four parameters was not easily modified to the long-run equilibrium with S&P 500 index futures.Empirical findings indicate that the volatility index, US dollar index, and unemployment rate have significantly negative relationships with S&P 500 index.This result suggests that the investors expect a decrease in S&P 500 index when VIX, US dollar index, and unemployment rate increase.However, the interest rate has a significant positive relationship with S&P 500 index, suggesting that S&P 500 index rises when the the interest rate increases, suggesting that there is an optimism about a future business boom, so that the S&P 500 index rises as a result.
Empirical results also prove that the factors chosen via the D-ANP are not significantly different from those obtained using the regression model, implying that S&P 500 investment decisions based on the D-ANP have similarly explanatory power to those obtained from the regression model.

GRA
For GRA, the GRC is computed to demonstrate the relationship between the ideal and the actual empirical findings.A multi-criteria problem is defined using a set of choices (x 1 , x 2 , . . ., x m ) with n criteria.The GRC, ξ k (x i , x j ), is expressed as ξ k x i , x j = ∆ min + ρ∆ max ∆ jik + ρ∆ max (9) where x i denotes a reference sequence, and x j represents a comparative sequence; ∆ jik is defined as the grey relational space, and ξ k (x i , x j ) is between 0 and 1.
where | .| denotes the absolute value and is the distinguishing coefficient (0 ≤ ≤ 1).Liu and Lin [2] reported that = 0.5 is normally applied.After obtaining the GRC, its mean value is often used as the GRG, γ(x i , x j ): where γ(x i , x j ) represents GRG for the i th experiment, and j shows the number of performance characteristics (taking value between 0 and 1), w k denotes the relative weight of performance characteristic k; and w 1 , w 2 , . . ., w n are usually satisfied as: n ∑ j=1 w j = 1 (14)

Conclusions
We combined the D-ANP with GRA to examine the key factors for investor trading in S&P 500 index futures and mutual relationships among key factors.We can draw the following conclusions.

1.
Thirteen experts picked five key factors out of 19 factors affecting investor trading in S&P 500 index futures.These key factors were the US dollar index, interest rate, ISM manufacturing PMI, volatility index, and unemployment rate.We found that the US dollar index is the most important among these five key factors.

2.
Previous studies concentrated on the explanatory power of technical indicators for S&P 500 index futures.Here, we found a weight for each key factor using the D-ANP, and we also considered various macroeconomic factors, which were found to have more explanatory power than those of technical factors found in previous research papers.3.
The D-ANP results revealed that the interest rate and US dollar index have mutually causal relationships, while the US dollar index unilaterally affects ISM manufacturing PMI, volatility index, and unemployment rates.4.
The co-integration results showed that there were at least two co-integration relationships that existed among the six factors.We also found that the correction term to unemployment rate has a significantly negative effect at 1-lag period, and we found that the unemployment rate was easily modified to the long-run equilibrium.

5.
Granger causality test results confirmed some similar results obtained via the D-ANPs that the US dollar index, interest rate, and the ISM manufacturing PMI have major impacts on the S&P 500 index futures.6.
The general regression results also confirmed that four out of the five factors selected via the D-ANP (volatility index, US dollar index, interest rate, and unemployment rate) have strong explanatory power in forecasting S&P 500 index futures.7.
We used the GRA to examine the explanatory power of the 12 factors selected by the D-ANP for different S&P 500 sectors.Empirical results indicated that the 12 factors had the strongest explanatory power for S&P 500 Industrial sector and the least explanatory power for S&P 500 Utility sector. 8.
We applied the GRA to measure the explanatory power of the 12 factors selected via the D-ANP for three major US stock indexes futures.Empirical findings showed that the 12 factors had the strongest explanatory power in forecasting S&P 500 index futures, while the least explanatory power in forecasting NASDAQ 100 index futures.

3 .
Amendment of the Prototypical Structure

Figure 2 .
Figure 2. Amendment of the prototypical structure.

Figure 2 .
Figure 2. Amendment of the prototypical structure.

Stage 1 :
Model formation and problem arrangement; Stage 2: Pairwise comparison matrices and preference vectors; Stage 3: Supermatrix formulation; Stage 4: Select the best possible choices.

Stage 1 :
Model formation and problem arrangement; Stage 2: Pairwise comparison matrices and preference vectors; Stage 3: Supermatrix formulation; Stage 4: Select the best possible choices.

Figure 5 .
Figure 5.A causal diagram for the five core factors.

Table 1 .
Academic and professional background for 13 experts.
* M denotes Master, B denotes Bachelor, P denotes Ph.D.

Table 1 .
Academic and professional background for 13 experts.
* M denotes Master, B denotes Bachelor, P denotes Ph.D.

Table 2 .
Amendment of the prototypical structure.

Table 2 .
Amendment of the prototypical structure.

Table 3 .
The scores of importance for the first-round questionnaire.

Table 4 .
Importance scores for second-round questionnaire.

Table 5 .
Measurement scores for the influence of relationships.

Table 6 .
Direct impact matrix for 12 factors.

Table 7 .
The normalized direct impact matrix for 12 factors.

Table 8
indicates the total impact matrix for 12 factors.

Table 8 .
The total impact matrix for 12 factors.

Table 9 .
Prominence and relation for 12 factors.

Table 12 .
Relative weights and rankings according to the ANP method.

Table 14
summarized the definition of six variable.

Table 14 .
Definition of variables.

Table 15 .
Summary of descriptive statistics.
4.3.4.Co-Integration Test ResultTable17denotes that at least two co-integration relationships exist among the six factors.

Table 17
denotes that at least two co-integration relationships exist among the six factors.

Table 17 .
Result for the co-integration test.The Lagged Period for Unemployment Rate and VECM Result Table18demonstrates that the correction error term to unemployment rate has a significantly negative effect at 1-lag period, where Schwartz Information Criteria (SIC) deals with the optimum lag length..

Table 18 .
The result for the lagged period for unemployment rate.
Note: * indicates the optimum lagged period based on Schwartz Information Criteria (SIC) rule.

Table 19 .
The adjustment speed.

Table 20
shows that the US dollar index unilaterally affects S&P 500 and VIX index; the interest rate unilaterally affects S&P 500, US dollar index, and VIX index; and the PMI unilaterally affects S&P 500 and interest rate.

Table 20 .
Granger causality test results.This study then chooses the top five factors selected by the D-ANP (i.e., US dollar index, interest rate, ISM manufacturing PMI, VIX, and unemployment rate) to be the independent variables, and the rate of return on S&P 500 index futures to be the dependent variable to establish a regression model.The sample period starts from 1 January 2006 to December 2014.The estimated results are summarized below: R1t = 0.005 − 0.813USDX 1t + 0.068IR 1t + 0.012PMI 1t − 0.14V IX 1t − 0.07UR 1t